Method and apparatus for providing enhanced pay per view in a video server employing a coarse-grained striping scheme

ABSTRACT

A method and apparatus are disclosed for providing enhanced pay per view in a video server. Specifically, the present invention periodically schedules a group of non pre-emptible tasks corresponding to videos in a video server having a predetermined number of processors, wherein each task begins at predetermined periods and has a set of sub-tasks separated by predetermined intervals. To schedule the group of tasks, the present invention divides the tasks into two groups according to whether they may be scheduled on a single processor. The present invention schedules each group separately. For the group of tasks not scheduleable on a single processor, the present invention determines a number of processors required to schedule such group and schedules such tasks to start at a predetermined time. For the group of tasks scheduleable on a single processor, the present invention determines whether such tasks are scheduleable on the available processors using an array of time slots. If the present invention determines that such group of tasks are not scheduleable on the available processors, then the present invention recursively partitions such group of tasks in subsets and re-performs the second determination of scheduleability. Recursive partitioning continues until the group of tasks is deemed scheduleable or no longer partitionable. In the latter case, the group of tasks is deemed not scheduleable.

RELATED APPLICATIONS

The subject matter of this application is related to U.S. patent application Ser. No. 08/624,013, entitled "Method and Apparatus for Providing Enhanced Pay Per View in a Video Server" filed concurrently herewith. The subject matter of this application is also related to the U.S. patent application Ser. No. 08/492,315, entitled "Coarse-Grained Disk Striping Method for Use in Video Server Environments", each of the above applications having at least one common inventor and a common assignee, the subject matter of each application being incorporated by reference herein.

FIELD OF THE INVENTION

The present invention relates generally to the field of video-on-demand services, and more particularly to video servers employing disk stripping methodology.

BACKGROUND OF THE INVENTION

In recent years, significant advances in both networking technology and technologies involving the digitization and compression of video have taken place. For example, it is now possible to transmit several gigabits of data per second over fiber optic networks and with compression standards like MPEG-1, the bandwidth required for transmitting relatively low. These advances have resulted in a host of new applications involving the transmission of video data over communications networks, such as video-on-demand, on-line tutorials, interactive television, etc.

Video servers are one of the key components necessary to provide the above applications. Depending on the application, the video servers may be required to store hundreds of video programs and concurrently transmit data for a few hundred to a few thousand videos to clients. As would be understood, the transmission rate is typically a fixed rate contingent upon the compression technique employed by the video server. For example, the transmission rate for MPEG-1 is approximately 1.5 Mbps.

Videos, for example movies and other on-demand programming, are transmitted from the random access memory (RAM) of the video server to the clients. However, due to the voluminous nature of video data (e.g., a hundred minute long MPEG-1 video requires approximately 1.125 GB of storage space) and the relatively high cost of RAM, storing videos in RAM is prohibitively expensive. A cost effective alternative manner for storing videos on a video server involves utilizing magnetic or optical disks instead of RAM. Video stored on disks, however, needs to be retrieved into RAM before it can be transmitted to the clients by the video server. Modern magnetic and optical disks, however, have limited storage capacity, e.g., 1 GB to 9 GB, and relatively low transfer rates for retrieving data from these disks to RAM, e.g., 30 Mbps to 60 Mbps. This limited storage capacity affects the number of videos that can be stored on the video server and, along with the low transfer rates, affects the number of videos that can be concurrently retrieved.

A naive storage scheme in which an entire video is stored on an arbitrarily chosen disk could result in disks with popular video programming being over-burdened with more requests than can be supported, while other disks with less popular video programs remain idle. Such a scheme results in an ineffective utilization of disk bandwidth. As would be understood, the term "disk bandwidth" refers to the amount of data which can be retrieved from a disk over a period of time. When data is not being retrieved from a disk, such as when the disk is idle or when a disk head is being positioned, disk bandwidth is not being utilized, and is thus considered wasted. Ineffective utilization of disk bandwidth adversely affects the number of concurrent streams a video server can support.

To utilize disk bandwidth more effectively, various schemes have been devised where the workload is distributed uniformly across multiple disks, i.e., videos are laid out on more than one disk. One popular method for storing videos across a plurality of disks is disk striping, a well known technique in which consecutive logical data units (referred to herein as "stripe units") are distributed across a plurality of individually accessible disks in a round-robin fashion. Disk stripping, in addition to distributing the workload uniformly across disks, also enables multiple concurrent streams of a video to be supported without having to replicate the video.

Outstanding requests for videos are generally serviced by the video server in the order in which they were received, i.e., first-in-first-out (FIFO). Where the number of concurrent requests is less than or not much greater than the number of concurrent streams that can be supported by the server, good overall response times to all outstanding requests are possible. In video-on-demand (VOD) environments, however, where the number concurrent requests, typically, far exceeds the number of concurrent streams that can be supported by the server, good overall response times are not possible for all outstanding requests using FIFO.

To provide better overall response times, a paradigm known as enhanced pay-per-view (EPPV) was adopted by VOD environments, such as cable and broadcasting companies. Utilizing the enhanced pay per view paradigm, video servers retrieve and transmit video streams to clients at fixed intervals or periods. Under this paradigm, the average response time for a request is half of the fixed interval and the worst case response time for a request would be the fixed interval. Furthermore, by retrieving popular videos more frequently, and less popular video less frequently, better overall average response times could be achieved. Finally, by informing the clients about the periods and the exact times at which videos are retrieved, zero response times can be provided.

Although a set of videos are scheduleable on a video server employing the EPPV paradigm, determining an exact schedule for which new streams of videos are to begin can be difficult, particularly when the periods and computation times, i.e., time required to transmit a video or segment, are arbitrary. The goal is to schedule the set of videos such that the number of streams scheduled to be transmitted concurrently does not exceed the maximum number of concurrent streams supportable by the video server. The complexity of scheduling videos in an EPPV paradigm is NP-hard--wherein the complexity of scheduling periodic videos increases exponentially as the number of videos being scheduled and the number of processors by which the videos are transmitted increases. Accordingly, there is a need for a method and apparatus which can effectively schedule videos periodically on a video server employing the EPPV paradigm.

SUMMARY OF THE INVENTION

The present invention sets forth a method for providing enhanced pay per view in a video server. Specifically, the present method periodically schedules a group G of non pre-emptible tasks T_(i) corresponding to videos in a video server having p number of processors, wherein each task T_(i) begins at predetermined periods P_(i) and has w_(i) number of sub-tasks separated by intervals F. Four procedures are utilized by the present method to schedule the tasks T_(i) in the group G. The first procedure splits a group G of tasks contained in a service list into two sub-groups G1 and G2 of tasks according to whether the tasks are scheduleable on a single processor. The first procedure subsequently attempts to schedule both sub-groups G1 and G2 separately.

For the sub-group of tasks not scheduleable on a single processor, i.e., sub-group G1, the first procedure schedules such tasks on p' number of processors. For the sub-group of tasks scheduleable on a single processor, i.e., sub-group G2, the first procedure calls upon a second procedure to further split the sub-group G2 into subsets G2-y such that there is one subset G2-y for each of the remaining p" number of processors.

For each subset G2-y, the first procedure calls a third procedure to determine scheduleability. The third procedure uses a fourth procedure to assist the third procedure in determining scheduleability of a subset G2-y. If the fourth procedure determines that a subset G2-y is not scheduleable and the third procedure determines that the same subset G2-y can be further partitioned, then the third procedure calls upon the second procedure to further divide the subsets G2-y into sub-subsets S_(v).

After the subset G2-y has been partitioned, the third procedure calls itself to re-determine scheduleability. The present invention will recursively partition a sub-group, subset, etc. of tasks until the second scheduling scheme determines whether the sub-group, subset, etc. of tasks is scheduleable or not partitionable any further.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the present invention, reference may be had to the following description of exemplary embodiments thereof, considered in conjunction with the accompanying drawings, in which:

FIG. 1 depicts a video server for providing video-on-demand (also referred to herein as "VOD") services in accordance with the present invention;

FIG. 2 depicts an exemplary illustration of disk striping;

FIG. 3 depicts an exemplary fine-grained striping scheme for storing and retrieving videos V_(i) ;

FIG. 4 depicts an exemplary first coarse-grained striping scheme for storing and retrieving videos V_(i) ;

FIG. 5 depicts an exemplary second coarse-grained striping scheme for storing and retrieving videos V_(i) ;

FIGS. 6a to 6e depict an exemplary flowchart of a procedure for scheduling tasks with computation times C_(i) and periods P_(i) in a video server;

FIG. 7 depict an exemplary diagram proving the validity of Theorem 1;

FIGS. 8a to 8d depict an exemplary flowchart of a procedure for scheduling a group of tasks in FIGS. 6a to 6e with computation times greater than its periods;

FIGS. 9a to 9d depict an exemplary flowchart of a procedure for partitioning a group of tasks;

FIGS. 10a to 10c depict an exemplary flowchart of a procedure for scheduling tasks having sub-tasks, each of unit computation time, at periods P_(h) and an interval F;

FIGS. 11a to 11b depict an exemplary flowchart of a procedure for scheduling a group of tasks in FIGS. 10a to 10c which are scheduleable on less than one processor; and

FIGS. 12a to 12c depict an exemplary flowchart of a procedure for determining scheduleability of tasks in FIGS. 10a to 10c using an array of time slots.

DETAILED DESCRIPTION

Referring to FIG. 1, there is illustrated a video server 10 for providing video-on-demand (VOD) services in accordance with the present invention. The video server 10 is a computer system comprising processors 11-1 to 11-p, a RAM buffer memory 12 of size D and a data storage unit 13. The data storage unit 13 comprises a plurality of disks 13-1 to 13-m for the storage of videos V_(i), where i=1, . . . , N denotes a specific video (program), or a group of concatenated videos V_(i) (hereinafter referred to as a "super-video V_(h) ", where h=1, . . . , P denotes a specific super-video), which are preferably in compressed form. The data storage unit 13 further comprises m disk heads (not shown) for retrieving the videos V_(i) from the disks 13-1 to 13-m to the RAM buffer memory 12. Note that the value of p, which is the number of processors, depends on the size D of the RAM buffer memory, the m number of disks and the amount of data that is being retrieved. The manner in which the value of p is determined will be described herein.

The processors 11-1 to 11-p are operative to transmit the videos V_(i) across a high-speed network 14 at a predetermined rate, denoted herein as r_(disp), to one or more recipients or clients 15. However, before the videos V_(i) can be transmitted, the videos V_(i) must first be retrieved from the disks 13-1 to 13-m into the RAM buffer memory 12 (from which the videos V_(i) are eventually transmitted). The continuous transfer of a video V_(i) from disks 13-1 to 13-m to the RAM buffer memory 12 (or from the RAM buffer memory 12 to the clients) at the rate r_(disp) is referred to herein as a stream. The number of videos V_(i) that can be retrieved concurrently from the disks 13-1 to 13-m, i.e., concurrent streams, is limited by the size D of the RAM buffer memory 12--the RAM buffer memory size is directly proportional to the number of retrievable concurrent streams.

Video servers operative to retrieve or support a large number of concurrent streams provide better overall response times to video requests by clients. Ideally, a video server would have a RAM buffer memory of sufficient size D to support every concurrent client request immediately upon receipt of the request. In such an ideal situation, the video server is described as having zero response times to client requests. However, in VOD environments, where the number of concurrent requests per video server are large, the size of the RAM buffer memory required to service the requests immediately would impose a prohibitively high cost on the video server. In such an environment, the number of requests far exceeds the capacity of the video server to transmit concurrent streams, thus adversely affecting overall response times. To provide better overall response times, VOD environments such as cable and broadcast companies adopted a very effective paradigm, known as enhanced pay per view or EPPV, for retrieving and transmitting videos V_(i) to clients periodically.

Enhanced Pay Per View

Enhanced pay per view involves retrieving and transmitting new streams of video V_(i) to clients at a period P_(i), where P_(i) is expressed in terms of time intervals referred to as rounds For example, starting with a certain round r_(i), data retrieval for new streams of video V_(i) is begun during rounds r_(i), r_(i) +P_(i), r_(i) +2·P_(i), etc. However, depending on the period P_(i) and the size of the available RAM buffer memory, it may not always be possible to retrieve data for the videos V_(i) at period P_(i). The problem of determining whether videos may be retrieved periodically by a video server is the same as scheduling periodic tasks, denoted herein as T_(i), on a multiprocessor. See J. A. Stankovic and K. Ramamritham, "Hard Real-Time Systems" published in IEEE Computer Society Press, Los Alamitos, Calif., 1988. For the purposes of this application, a task T_(i) (or task T_(h)) corresponds to the job of retrieving all data belonging to a stream of video V_(i) (or super-video V_(h)) on a single disk.

Note that EPPV may be unsuitable for environments in which the number of concurrent requests is not much larger than the number of concurrent streams supportable by the video server, and in which the requested videos are random, making it difficult to determine periods for the various videos.

Disk Striping

In accordance with the present invention, a novel method and apparatus is disclosed for providing enhanced pay per view in a video server employing a disk striping scheme. Disk striping involves spreading a video over multiple disks such that disk bandwidth is utilized more effectively. Specifically, disk striping is a technique in which consecutive logical data units (referred to herein as "stripe units") are distributed across a plurality of individually accessible disks in a round-robin fashion. Referring to FIG. 2, there is shown an exemplary illustration depicting a disk striping methodology. As shown in FIG. 2, a video 20 is divided into z number of stripe units 22-1 to 22-z of size su. Each stripe unit is denoted herein as V_(ij), where j=1, . . . , z denotes a specific stripe unit of video V_(i).

For illustration purposes, the present invention will be described with reference to particular fine-grained and coarse-grained disk striping schemes disclosed in a related patent application, U.S. Ser. No. 08/492,315, entitled "Coarse-Grained Disk Striping Method For Use In Video Server Environments." However, this should not be construed to limit the present hommoiy to the aforementioned particular disk striping schemes in any manner.

Fine-Grained Striping Scheme

Referring to FIG. 3, there is illustrated an exemplary fine-grained striping scheme for storing and retrieving videos V_(i). In this fine-grained striping scheme ("FGS scheme"), as shown in FIG. 3, stripe units V_(ij) 32-1 to 32-m belonging to one or more videos V_(i) are stored contiguously on disks 30-1 to 30-m in a round-robin fashion--stripe units V_(i),l to V_(i),m are stored on disks 30-1 to 30-m, respectively, stripe units V_(i),l+m to V_(i),2m are stored on disks 30-1 to 30-m, respectively, etc. Each group of m number of consecutive stripe units V_(ij) are stored in the same relative position on their respective disks--for example, stripe units V_(i),l to V_(i),m are stored on the first track of disks 30-1 to 30-m, respectively, stripe units V_(i),l+m to V_(i),2.m are stored on the second track of disks 30-1 to 30-m, respectively, etc. Each video V_(i) preferably has a length l_(i) which is a multiple of m such that, for each disk 30-1 to 30-m, the number of stripe units belonging to the same video V_(i), denoted herein as w_(i), is the same for each disk. This can be achieved by appending advertisements or padding videos at the end. The group of stripe units on a disk belonging to the same video is referred to herein as a block, thus each block has w_(i) stripe units belonging to the video V_(i).

In this striping scheme, data for each stream of video V_(i) is retrieved from disks 13-1 to 13-m using all m disk heads simultaneously. Specifically, each disk head retrieves a stripe unit of size su from the same relative position on their respective disks for each stream over a series of rounds denoted herein as r_(i). Thus, the amount of data retrieved during a round r_(i) for each stream constitutes a group of m number of consecutive stripe units belonging to the video V_(i). Each group of m number of stripe units constitutes a size d portion of the video V_(i) --in other words, m·su=d. Note that the number of rounds for which data is retrieved for a stream of video ##EQU1## where l_(i) corresponds to the length of video V_(i). Accordingly, each stream of video V_(i) can be viewed as comprising ##EQU2## number of size d portions.

When data retrieval is begun for a new stream of video V_(i), a buffer of size 2.d is allocated. This allows size d portions of each new stream to be retrieved and transmitted concurrently. Note however that data transmission for a new stream does not begin until the entire first size d portion of the video V_(i) is retrieved into the allocated buffer. For example, if round l_(i) is the time interval in which the first size d portion of a video V_(i) is retrieved into the allocated buffer, then transmission to the clients of the first size d portion at the rate r_(disp) is not begun until round 2_(i). The time it takes to transmit a size d portion of data corresponds to the duration of the round r_(i) --in other words, the time interval of round r_(i) is ##EQU3## Additionally, in round 2_(i), during the transmission of the first size d portion of video V_(i), the second size d portion of video V_(i) is being retrieved into the allocated buffer.

Generally, for each round r_(i), the next size d portion of a video V_(i) is retrieved into the allocated buffer and the preceding size d portion of the same video V_(i), which is currently residing in the allocated buffer, is transmitted from the allocated buffer. This action of simultaneously retrieving and transmitting size d portions of data continues uninterrupted until the entire video V_(i) is retrieved and transmitted. Note that data retrieval of the next size d portion of video V_(i) must be completed at the same time or before transmission of the preceding size d portion of video V_(i) is completed--i.e., the time to retrieve a size d portion must be equal to or less than the time to transmit a size d portion. In other words, data retrieval must be completed with the round r_(i). The reason for this requirement is the need to ensure that data for every video is transmitted to the clients as a continuous stream at the rate r_(disp). Failure to complete data retrieval within each round r_(i) could result in an interruption in the video stream.

The size increments at which data is being retrieved for each stream of video V_(i) is uniform, i.e., value of d, and is preferably computed based on the size D of the RAM buffer memory and the m number of disks on which the video V_(i) is striped across such that the number of concurrent streams that can be supported by the video server is maximized. For the present FGS scheme, the value of d is ##EQU4## is the maximum value of d, t_(seek) is the time it takes for a disk head to be positioned on the track containing the data, t_(rot) is the time it takes the disk to rotate so the disk head is positioned directly above the data, and t_(settle) is the time it takes for the disk head to adjust to the desired track. Otherwise the value of d is ##EQU5##

Maximizing the number of concurrent streams supportable by the video server also depends on the number of processors in a video server. As mentioned earlier, each processor 11-1 to 11-p is operative to transmit data at the rate r_(disp). Since a duration of a round r_(i) is ##EQU6## then each processor 11-1 to 11-p can only transmit one size d portion of data per round r_(i). Thus, the maximum number of concurrent streams supportable by a video server, denoted herein as q_(max), is limited by the p number of processors. Recall that in the present invention the value of p is computed based on the size D of the RAM buffer memory, the m number of disks and the amount of data that is being retrieved, i.e., size d portions, such that the number of concurrent streams supportable by the video server is maximized. For the present FGS scheme, the number of processors is ##EQU7## which is the minimum amount needed to maximized the number of concurrent streams supportable by the video server. In other words, p=q_(max)

Data for each concurrent stream of video V_(i) is retrieved from the disks according to a service list assembled by the video server every round from outstanding client requests. The service list is an index of video V_(i) streams for which data is currently being retrieved. More specifically, the service list indicates the relative position of the stripe units on the disks for each stream of video V_(i) in which data is currently being retrieved. Note that a service list may contain entries indicating data retrieval of different stripe units belonging to the same video V_(i) if they are for different streams. Further note that data retrieval for each and every video V_(i) stream on the service list must be completed before the completion of the round otherwise there may be an interruption in the video V_(i) stream. In the present FGS scheme, only one service list is required to instruct all m disk heads of the proper stripe units to retrieve, since each disk head, as discussed earlier, simultaneously retrieves the same relatively positioned stripe unit on their respective disks.

The number of entries on the service list, denoted herein as q, reflect the number of concurrent streams being retrieved. In light of such, the number of entries on the service list must never exceed the maximum number of concurrent streams that can be supported by the video server, i.e., q≦q_(max). The mere fact that the q number of videos on the service list does not exceed q_(max) does not guarantee that the videos V_(i) listed in the service list are scheduleable on p number of processors. Scheduling conflicts typically arise as a result of the individual periods P_(i) in which new streams of videos V_(i) are begun.

In the FGS scheme, the problem of scheduling data retrieval for videos V_(i) periodically is the same as scheduling non pre-emptible tasks T_(i), . . . , T_(n) having periods P_(l), . . . , P_(n) and computation times C_(i) on p processors. A task T_(i), as mentioned earlier, corresponds to the job of retrieving, for a stream of video V_(i), all data belonging to the video V_(i) on a single disk. Applying this definition to the present fine-grained striping scheme, a task T_(i) would consist of retrieving a block of w_(i) number of stripe units belonging to the same video V_(i) from a disk--for example, retrieving the stripe units V_(i),l to V_(i),l+w.m from disk 30-1. The time needed to complete the task T_(i) is referred to herein as computation time C_(i) or run time, which is expressed in terms of rounds. For the FGS scheme, the computation time C_(i) is equal to the number of rounds needed to retrieve the entire block of stripe units belonging to the video V_(i) on a disk, i.e., ##EQU8## Note that a simpler problem of determining whether periodic tasks with unit computation times are scheduleable on a single processor has been shown to be NP-hard by S. Baruah, et. al. in "International Computer Symposium, Taiwan," pages 315-320, published 1990. The term "NP-hard" is well-known in the art and refers to a complexity of a problem.

First Coarse-Grained Striping Scheme

Referring to FIG. 4, there is illustrated an exemplary first coarse-grained striping scheme (also referred to herein as "first CGS scheme") for storing and retrieving videos V_(i). The first coarse-grained striping scheme is similar in every manner to the FGS except as noted herein. As shown in FIG. 4, stripe units V_(hj) 42-1 to 42-m belonging to one or more super-videos V_(h), i.e., a group of concatenated videos V_(i), are stored contiguously on disks 40-1 to 40-m in a round-robin fashion. In contrast to the FGS scheme, the size su of the stripe units in the first CGS scheme are larger and preferably equal to d. Every video V_(i) comprising each super-video V_(h) preferably have a length l_(i) that is a multiple of d and m such that each disk has a block with w_(h) number of stripe units belonging to the super-video V_(h).

Coarse-grained striping differs fundamentally from fine-grained striping in that coarse-grained striping schemes typically retrieve data for each stream of super-video V_(h) using one disk head at a time. The reason is that, in the present scheme, each stripe unit has a size su=d. This is in contrast to the FGS scheme, where each stripe unit has a size ##EQU9## In subsequent rounds r_(h), of the first CGS scheme, successive stripe units belonging to the same super-videos V_(h) are retrieved from successive disks by its respective disk head--for example, stripe unit V_(h),l 42-1 is retrieved from disk 40-1 in the first round, stripe unit V_(h),2 42-2 is retrieved from disk 40-2 in the second round, etc. Unlike the FGS scheme, adjacent stripe units on a disk belonging to the same super-videos V_(h) are retrieved at intervals of F number of rounds apart in the first CGS scheme. Specifically, in the first CGS scheme, adjacent stripe units on a disk are retrieved at intervals of m number of rounds since consecutive stripe units are stored contiguously on the m disks in a round-robin fashion--for example, stripe unit 42-1 is retrieved from disk 40-1 in the first round, and the next stripe unit on disk 40-1, i.e., stripe unit 42-1+m, is retrieved in round 1+m.

For every disk head in the present scheme, a separate service list is provided by the video server, thus there are m number of service lists. Each entry on these service lists indicates a size d portion of data being retrieved from a single disk. Contrast with the FGS scheme, where each entry represents a size ##EQU10## portion of data being retrieved per disk. Since each entry indicates retrieval for one size d portion of data, a processor is required for each entry. If q_(disk-max) is the maximum number of entries on a service list, then q_(disk-max).m is the maximum number of concurrent streams the video server is operative to support. In other words, the minimum number of processors required for the first CGS scheme such that the number of concurrent streams supportable by the video server is maximized is p=q_(disk-max).m. The manner of determining the value of p is disclosed herein. The number of entries in a service list is denoted herein as q_(disk).

In the first CGS scheme, at the end of each round r_(h), the service list for every disk except the first disk is set to the service list of its preceding disk--for example, the service list for disk 30-1 in round 1 becomes the service list for disk 30-2 in round 2. The reason for this is that the successive size d portions of video to be retrieved in the next round is in the same relative position on the successive disk as the preceding size d portion was on the preceding disk. The only disk that is provided with a brand new service list every round is the first disk. Accordingly, if the super-videos V_(h) entries contained in the service list for the first disk is scheduleable for data retrieval, then the same super-videos V_(h) can be scheduled for data retrieval for the other disks in the successive rounds r_(h).

For the first coarse-grained striping scheme, the value of d is d_(calc) if .left brkt-bot.q_(disk-max) .right brkt-bot..m is greater than ##EQU11## is the maximum value of d, and ##EQU12## is the maximum number of streams that can be supported by each disk per round for d=d_(calc). Otherwise the value of d is d.sub..left brkt-top.calc.right brkt-top., which is the minimum value of d needed to support .left brkt-top.q_(disk-max) .right brkt-top. streams from a disk per round. Since there are m number of disks, the maximum number of size d portions of super-videos V_(h) that can be retrieved is q_(disk-max) ·m. Like the FGS scheme, the number of processors required for the first CGS scheme is computed based on the size D of the RAM buffer memory, the m number of disks and the maximum amount of data that can be retrieved. For the first CGS scheme, the value of p is ##EQU13##

In the first CGS scheme, the problem of retrieving data for the super-videos V_(h) periodically is the same as that of scheduling tasks T_(h), i.e., T_(l), . . . , T_(R), having w_(h) non-consecutive stripe units on p processors. Each task T_(h) has w_(h) number of sub-tasks, and each sub-task has a unit computation time. An interval of F number of rounds is interposed between adjacent sub-tasks belonging to the same task T_(h). Thus, the computation time C_(h) for task T_(h) is w_(h) number of rounds plus (w_(h) -1)·m number of rounds, i.e., (2w_(h) -1)·m number of rounds. Note that a simpler case of scheduling tasks having single sub-tasks on a multiprocessor was shown to be NP-hard by S. Baruah, et. al., supra.

Second Coarse-Grained Striping Scheme

Referring to FIG. 5, there is illustrated an exemplary second coarse-grained striping scheme (also referred to herein as "second CGS scheme") for storing and retrieving videos V_(i). The second CGS scheme is similar in every manner to the first CGS scheme except as noted herein. In contrast to the first coarse-grained striping scheme, blocks 51-1 to 52-m comprising w_(h) number of consecutive stripe units_(hj) belonging to one or more super-video V_(h) are stored contiguously on disks 50-1 to 50-m in a round-robin fashion, as shown in FIG. 5, where ##EQU14## and l_(h) corresponds to the length of super-video V_(h). In the second CGS scheme, the value of w_(h) is identical for each and every block of super-video V_(h), and the length l_(h) of each super-video V_(h) is a multiple of w_(h) such that each disk on which a block of super-video V_(h) is stored has a block of w_(h) number of consecutive stripe units for each super-video V_(h). For every super-video V_(h), the first w_(h), stripe units are stored on disk 50-1. Successive groups of w_(h) consecutive stripe units are stored on successive disks.

Similar to the first coarse-grained striping scheme, the second coarse-grained striping scheme retrieves size d stripe units of the super-video V_(h) from the same disk using one disk head, starting at some round r_(h) and at subsequent intervals of P_(h) rounds. Unlike the first coarse-grained striping scheme, and more like the fine-grained striping scheme, the second coarse-grained scheme continuously retrieves data from the same disk for the same video V_(h) stream in consecutive rounds r_(h) until the entire block of w_(h) number of stripe units are retrieved from that disk--for example, stripe units 52-1 to 52-w_(h) are retrieved from disk 50-1 in the first w_(h) number of rounds, stripe units 52-1+w_(h) to 52-2.w_(h) are retrieved from disk 50-2 in the next w_(h) number of rounds, etc. Thus, there are no intervals of F number of rounds interposed between adjacent sub-tasks belonging to the same task T_(h).

Since there are no intervals interposed between consecutive sub-tasks of the same task T_(h), the problem of scheduling videos periodically in the second CGS scheme is similar to the problem of scheduling videos periodically in the FGS scheme--that is, scheduling non pre-emptible tasks T_(l), . . . , T_(R) having periods P_(l), . . . , P_(R) and computation times C_(h). For the second CGS scheme, each task T_(l) consist of retrieving w_(h) number of consecutive stripe units, thus there are w_(h) number of sub-tasks. The computation times C_(h) is equal to w_(h) number of rounds.

Scheduling Schemes

In the following subsections, scheduling schemes are presented for the two basic scheduling problems that arise when retrieving data for videos periodically on multiple processors. Generally, the scheduling schemes involve determining whether the tasks are scheduleable and, if they are determined to be scheduleable, scheduling start times for the tasks.

First Scheduling Scheme

The first scheduling scheme addresses the scheduling problem presented by the FGS and second CGS schemes in a video server employing the EPPV paradigm. Specifically, the first scheduling scheme provides a valid method for scheduling non pre-emptible tasks T_(l), i.e., T_(l), . . . , T_(n) or T_(l), . . . , T_(R), having periods P_(l), i.e., P_(l), . . . , P_(n) or P_(l), . . . , P_(R), and computation times C, (or C_(h)) on p number of processors. For ease of discussion, the first scheduling scheme is discussed with reference to the FGS scheme. However, such scheduling scheme should be construed to apply equally to the second CGS except as noted herein.

For the FGS scheme, the first scheduling scheme determines whether a video server having p number of processors is operative to service the group of videos contained in the one service list. Thus, all the videos on the service list must be scheduleable onp number of processors. Contrast this with the second CGS scheme where there are m number of service lists. As applied to the second CGS scheme, the first scheduling scheme determines whether a video server having p number of processors is operative to service the videos contained in the service list belonging to the first disk on p/m number of processors. Accordingly, for the second CGS scheme, the first scheduling scheme determines whether the videos on the service list belonging to the first disk are scheduleable on p/m number of processors.

The first scheduling scheme utilizes three procedures to address the scheduling problem presented by the FGS and second CGS schemes. The first procedure, referred to herein as SCHEME₋₋ 1, splits a group G of tasks into two sub-groups G1 and 2 of tasks according to whether the tasks are scheduleable on less than one processor. A task is scheduleable on less than one processor if one or more other tasks may be scheduled with it on the same processor. SCHEME₋₋ 1 subsequently attempts to schedule both sub-groups separately. For the sub-group of tasks not scheduleable on less than one processor, i.e., sub-group G1, SCHEME₋₋ 1 schedules such tasks on p' number of processors. Note that p=p'+p", where p is the number of processors in the video server. For the sub-group of tasks scheduleable on less than one processor, i.e., sub-group G2, SCHEME₋₋ 1 calls upon a second procedure, referred to herein as PARTITION, to further split the sub-group G2 into p" number of subsets G2-y such that there is one subset G2-y for each of the remaining p" number of processors. Subsequently, SCHEME₋₋ 1 makes a first determination whether each subset G2-y is scheduleable on one processor. If a subset G2-y is scheduleable on one processor, SCHEME₋₋ 1 schedules the tasks in the subset G2-y. Otherwise SCHEME₋₋ 1 calls upon a third procedure, referred to herein as SCHEDULE₋₋ TASKS₋₋ 1, to make a second determination of scheduleability for the subsets G2-y which were determined not to be scheduleable by the first determination. If SCHEDULE₋₋ TASKS₋₋ 1 determines that a subset is not scheduleable and that such subset can be further partitioned, then SCHEDULE₋₋ TASKS₋₋ 1 calls upon PARTITION to further divide the subset G2-y into sub-subsets S_(v). After a subset G2-y has been divided into sub-subsets S_(v), SCHEDULE₋₋ TASKS₋₋ 1 calls itself to re-make the second determination of scheduleability for every sub-subset S_(v) belonging to the partitioned subset G2-y. This technique is known as recursive partitioning. The first scheduling scheme will continue to recursively partition a sub-group, subset, sub-subset, etc. of tasks until the first scheduling scheme determines whether the sub-group, subset, sub-subset, etc. of tasks is scheduleable or not partitionable any further.

Referring to FIGS. 6a to 6e, there is illustrated an exemplary flowchart of the procedure SCHEME₋₋ 1 for scheduling a group of videos contained in a service list for a video server employing a FGS scheme, i.e., scheduling non pre-emptible tasks T_(l), . . . , T_(N) with periods P_(l), . . . , P_(N) and computation times C_(l), . . . , _(N). In step 600, as shown in FIG. 6a, data corresponding to tasks T_(i) in group G, where i=1, . . . , N, and a value of p are read into the RAM buffer memory of the video server. Data corresponding to group G includes tasks T_(i), periods P_(i) and computation times C_(i), i.e., {T_(i),P_(i),C_(i) }. Note that for the second CGS scheme, the value of p read into the RAM buffer memory will be p/m. In step 605, the tasks T_(i) are categorized into sub-groups G1 and 2 according to their corresponding periods P_(i) and computation times C_(i). Tasks T_(i) having computation times C_(i) that are greater than or equal to its periods P_(i) are not scheduleable on less than one processor because these tasks either monopolize an entire processor or will conflict with itself, i.e., two or more streams of the task will require the services of the processor at the same time. These tasks are placed in sub-group G1 and denoted herein as tasks T_(x), i.e., T_(x) ε G1. Tasks T_(i) having computation times C_(i) that are less than its periods P_(i) are scheduleable on less than one processor. These tasks are placed in sub-group G2 and denoted herein as tasks T_(y), i.e., T_(y) ε G2.

Example 1 is provided to illustrate the first scheduling scheme. The following value of p and data corresponding to tasks T_(i) in a service list are read into the RAM buffer memory in step 600:p=6, {T_(l), 4,8}, {T₂,4,1}, {T₃,4,1}, {T₄,3,7}, {T₅,6,1} and {T₆,6,1}. Tasks T₁, and T₄ are categorized as belonging to sub-group G1, i.e., G1={T₁,T₄ }, and tasks T₂, T₃, T₅ and T₆ are categorized as belonging t sub-group G2, i.e., G2={T₂, T₃, T₅, T₆ }, in step 605.

In steps 610 to 636, SCHEME₋₋ 1 schedules the tasks T_(x) belonging to the sub-group G1 on p' number of processors, where p' is the number of processors required to service all the tasks T_(x) ε G1. In step 610, the value of p' is initially set to zero. A loop 620 is initiated, in step 620, for every task T_(x) ε G1. Specifically, the loop 620 determines p_(x) number of processors required by the current task T_(x) in the loop 620 using the equation ##EQU15## in step 625, adds the current value of p_(x) to p' to progressively determine the total number of processors required by sub-group G1, in step 630, and schedules the current task T_(x) to start at a predetermined time, which preferably is time zero (expressed in terms of a round), i.e., ST_(x) =0, in step 635. The loop 620 ends when processor requirements and start times have been determined for every task T_(x) ε G1. Applying the steps in loop 620 to example 1, the following was determined: tasks T₁ and T₄ require two and three processors, respectively, and have been scheduled to start at time zero. Accordingly, five processors are reserved for sub-group G1, i.e.,p'=5.

Upon completing loop 620 for every task T_(x) or if there are no tasks T_(x) belonging to sub-group G1, SCHEME₋₋ 1 proceeds to step 645 where it determines the p" number of processors available for scheduling the tasks T_(y) ε G2, as shown in FIG. 6b. In step 646, if p" is less than zero, then the total number of processors required to service tasks T_(x) ε G1 is insufficient and SCHEME₋₋ 1 proceeds to step 647 where it returns a cannot₋₋ be₋₋ scheduled statement before continuing to step 685 where SCHEME₋₋ 1 terminates. Otherwise SCHEME₋₋ 1 proceeds to step 648 where it determines whether there are any tasks T_(y) belonging to the sub-group G2. If there are none, SCHEME₋₋ 1 goes to step 648a where it is instructed to go to step 685 where it terminates. Otherwise, SCHEME₋₋ 1 continues to step 648b where it checks if there are any processors available for servicing the tasks T_(y) ε G2. If there are no processors available, SCHEME₋₋ 1 goes to step 648c where a cannot₋₋ be₋₋ scheduled statement is returned before continuing to step 685 where SCHEME₋₋ 1 terminates.

Otherwise, SCHEME₋₋ 1 proceeds to divide the tasks T_(y) ε G2 into subsets G2-y such that there is one subset G2-y for each available processor, where y=1, . . . , p". As shown in FIG. 6c, in step 649, SCHEME₋₋ 1 checks if the number of available processors for tasks T_(y) is one. If p" is one, then all tasks T_(y) ε G2 need to be scheduled on the one available processor. Subsequently, SCHEME₋₋ 1 continues to step 650 where subset G2-1 is set equal to sub-group G2 before proceeding to step 660. If p" is not one, SCHEME₋₋ 1 continues from step 649 to steps 651 and 655 where the procedure PARTITION is called to recursively partition the sub-group G2 into p" number of disjoint subsets G2-y. The procedure PARTITION will be described in greater detail herein. In example 1, the number of available processors for sub-group G2 is one, thus subset G2-1 is set equal to G2, i.e., (T₂,T₃,T₅,T₆ }, in step 650. Tasks in the subsets G2-y are denoted herein as T_(d), where d=1, . . . , n, i.e., T_(d) ε G2-y.

A loop 660 is initiated for every subset G2-y in step 660 to make a first determination of scheduleability, i.e., is each subset G2-y scheduleable on one processor. In step 670, the following theorem, denoted herein as Theorem 1, is applied in SCHEME-1: if C_(l) +. . . +C_(n) ≦gcd(P_(l) +. . . +P_(n)), then tasks T_(l), . . . , T_(n) are scheduleable on a single processor, where gcd(P_(l) +. . . +P_(n)) is the greatest common divisor of the periods P_(d), and the start times ST₁, . . . , ST_(n) for each task T_(l), . . . , T_(n) is determined by adding the computation times C_(d) of the earlier scheduled tasks T_(d), i.e., ##EQU16## where 0≦d'<d. Note that when d=1, the first scheduled task T₁ starts at time zero. If Theorem 1 is satisfied for a subset G2-y, then every task T_(d) in the subset G2-y can be scheduled at ##EQU17## where Q is an arbitrary positive integer, and each T_(d) will have reserved for its exclusive use one or more rounds within each time interval of length equal to the greatest common divisor (also referred to herein as "gcd interval"). Note that failure to satisfy Theorem 1 does not conclusively determine that the subset G2-y is not scheduleable on one processor.

Referring to FIG. 7, there is illustrated an exemplary diagram proving the validity of Theorem 1. Consider tasks T₈, T₉ and T₁₀ having periods P₈ =3, P₉ =3 and P₁₀ =6 and unit computation times, i.e., C₈ =C₉ =C₁₀ =1. The gcd(3,3,6) is three and the sum of the computation times is three, and thus Theorem 1 is satisfied. Using the equation ##EQU18## new streams of tasks T₈, T₉ and T₁₀ are scheduled to begin in rounds 0, 1 and 2, respectively. As shown in FIG. 7, a schedule 70 is created for servicing tasks T₈, T₉ and T₁₀ on a single processor. Specifically, the first, second and third rounds in each gcd interval are reserved for the exclusive use by tasks T₈, T₉ and T₁₀, respectively--rounds 0, 3, 6, 9, 12 and 15 are reserved to begin new streams corresponding to task T₈, rounds 1, 4, 7, 10, 13 and 16 are reserved to begin new streams corresponding to task T₉ and rounds 2, 8 and 14 are reserved to begin new streams corresponding to task T₁₀. Note that rounds 5, 11 and 17 are also reserved for the exclusive use by task T₁₀ although no new streams are to begin in those rounds. Accordingly, by reserving the rounds within each gcd interval, scheduling conflicts are averted. In example 1, the sum of the computation times is four and the greatest common divisor is two, thus Theorem 1 is not satisfied and SCHEME₋₋ 1 proceeds to step 676.

As shown in FIG. 6d, if Theorem 1 is satisfied for the current subset G2-y in the loop 660, SCHEME₋₋ 1 continues to steps 671 to 675 where a loop 672 determines the start times for each task T_(d) ε G2-y using the equation ##EQU19## before proceeding to step 681. Specifically, the loop 672 adds the computation times of the earlier tasks to determine a start time for the current task in the loop 672. Upon completion of the loop 672, SCHEME₁₃ 1 goes to step 681 where it determines whether the loop 660 is complete. If Theorem 1 is not satisfied for a subset G2-y, SCHEME₋₋ 1 goes to steps 676 and 680 and makes a second determination of scheduleability for the subset G2-y that did not satisfy Theorem 1. In step 676, as shown in FIG. 6e, SCHEME₋₋ 1 looks for a minimum value of CG_(G2-y) that satisfies the following two conditions: (1) C_(G2-y) is greater than or equal to any C_(d) ε G2-y, i.e., C_(G2-y) ≧C_(d) ; and (2) C_(G2-y) divides evenly into every P_(d) ε G2-y, i.e., (C_(G2-y) mod P_(d))=0. If these conditions are not satisfied, SCHEME₋₋ 1 returns a cannot₋₋ be₋₋ scheduled statement, in step 677, before terminating. Otherwise SCHEME₋₋ l proceeds to step 680 where the procedure SCHEDULE₋₋ TASKS₋₋ 1 is called to make a second determination of scheduleability for the subset G2-y that failed to satisfy Theorem 1. Applying the instructions in steps 676 and 680 to example 1, the minimum value C_(G2-1) is set equal to one and SCHEDULE₋₋ TASKS₋₋ 1 is called to make a second determination of scheduleability for the subset G2-1.

Referring to FIGS. 8a to 8d, there is illustrated an exemplary flowchart of the procedure SCHEDULE₋₋ TASKS₋₋ 1. As shown in FIG. 8a, SCHEDULE₋₋ TASKS₋₋ 1 reads data as tasks T_(e) in group G3 and C_(G3), in step 800. Thus, tasks T_(d) ε G2-y and C_(G2-y) from SCHEME₋₋ 1 are read by SCHEDULE₋₋ TASKS₋₋ 1 as T_(e) ε G3 and C_(G3), respectively. Note that these calls to SCHEDULE₋₋ TASKS₋₋ 1 from SCHEME₋₋ 1 for the subsets G2-y are referred to herein as first level invocations of SCHEDULE₋₋ TASKS₋₋ 1. In steps 805 to 824, SCHEDULE₋₋ TASKS₋₋ 1 makes a second determination of scheduleability by packing the tasks T_(e) into g number of bins B-x of size C_(G3), where x=1, . . . , g, according to its computation times C_(e). In step 805, the value of g is determined using the equation ##EQU20## and, in step 810, the tasks T_(e) are packed into the bins B-x. Preferably, in step 810, the tasks T_(e) are sorted in descending order according to computation times C_(e) and subsequently assigned, one task at a time in sorted order, to a bin B-x for which the sum of the computation times C_(e) of previously assigned tasks T_(e) in that bin is minimum. An (x,a) pair indicates the location of a specific task T_(e) after it has been packed, where x is the bin number and a is the distance, expressed in terms of computation time, the task T_(e) is from the bottom of bin x. Applying steps 805 and 810 to example 1, the value of g is set to two and bin B-1 will contain, from bottom to top, tasks T₂ and T₅ and bin B-2 will contain, from bottom to top, tasks T₃ and T₆.

In step 820, each bin is checked to determine whether the sum of the computation times C_(e) for the tasks T_(e) assigned to the bin B-x is less than or equal to the value of C_(G3). If the condition in step 820 is satisfied for each bin B-x, then the tasks T_(e) belonging to group G3 will not conflict with each other and are scheduleable. In such a case, SCHEDULE₋₋ TASKS₋₋ 1 proceeds to steps 821 to 824, as shown in FIG. 8b, where the start times ST_(e) are determined for each task T_(e) ε G3 in a loop 821. Specifically, in step 822, the start time ST_(e) for a current task T_(e) in the loop 821 is calculated according to its (x,a) pair using the equation ST_(e) =(C_(G3) ·x)+a. Upon determining the start times ST_(e) for each task T_(e), the start times are returned to the calling program, in step 824. Note that the calling program is either SCHEME₋₋ 1 or a previous level invocation of SCHEDULE₋₋ TASKS₋₋ 1, as will be explained herein.

A later level invocation of SCHEDULE₋₋ TASKS₋₋ 1 may occur for the tasks T_(e) ε G3 if the condition in step 820 is not satisfied. In such a case, SCHEDULE₋₋ TASKS₋₋ 1 tries to recursively partition the group G3 into subsets S_(v), where v=1, . . . , g, and then tries to schedule the tasks in each of the subsets S_(v) by calling SCHEDULE₋₋ TASKS₋₋ 1 for each subset S_(v). For a group of tasks T_(e) ε G3 that do not satisfy the condition in step 820, SCHEDULE₋₋ TASKS₋₋ G2 proceeds to step 825 where it determines whether the group G3 can be recursively partitioned any further. If the value of g is one, then the group G3 cannot be recursively partitioned any further and, in step 826, a cannot₋₋ be₋₋ scheduled statement is returned to the calling program. If the value of g is not one, then the data in group G3 is transposed into group G3, in step 830, as shown in FIG. 8c, i.e., from {T_(e),P_(e),C_(e) } to {T_(e),P_(e),C_(e) } where ##EQU21## so it may be recursively partition into subset S_(v) by a call to the procedure PARTITION in step 835. In example 1, the sums of the computation times C₂ and C₅ for bin B-1 and computation times C₃ and C₆ for bin B-2 are both two, which is greater than C_(G3), which is one, thus the condition in step 820 is not satisfied. Accordingly, the data corresponding to tasks T_(e) in example 1 are transposed in step 830 before being partitioned. The transposed data is as follows: ##EQU22##

Upon returning from the call to PARTITION, SCHEDULE₋₋ TASKS₋₋ 1 attempts to schedule each subset S_(v) using a later level invocation of itself in steps 840 and 845. The first group of calls by SCHEDULE₋₋ TASKS₋₋ 1 to itself for each subset S_(v) is referred to herein as a second level invocation of SCHEDULE₋₋ TASKS₋₋ 1. Note that in the second level invocation of SCHEDULE₋₋ TASKS₋₋ 1, data corresponding to tasks T_(e) ε S_(v) are read as tasks T_(e) ε G3 by the later level invocation of SCHEDULE₋₋ TASKS₋₋ 1. A second level invocation of SCHEDULE₋₋ TASKS₋₋ 1 may lead to a third level invocation of SCHEDULE₋₋ TASKS₋₋ 1, which may lead to a fourth level invocation, etc., until either a cannot₋₋ be₋₋ scheduled statement or start times are returned to the previous level invocation of SCHEDULE₋₋ TASKS₋₋ 1. Recall that a level one invocation occurs when SCHEDULE₋₋ TASKS₋₋ 1 is called by SCHEME₋₋ 1 in step 680. In step 850, SCHEDULE₋₋ TASKS₋₋ 1 determines if a cannot₋₋ be₋₋ scheduled statement was returned by any level invocation of SCHEDULE₋₋ TASKS₋₋ 1. If such a statement was returned, the tasks T_(y) ε G2 are deemed not scheduleable on p" number of processors and SCHEDULE₋₋ TASKS₋₋ 1 returns a cannot₋₋ be₋₋ scheduled statement to its calling program in step 851.

If start times are returned to a previous level invocation of SCHEDULE₋₋ TASKS₋₋ 1 in step 845 by a later level invocation of SCHEDULE₋₋ TASKS₋₋ 1, then the start times are merged before being returned to the calling program in step 860, as shown in FIG. 8d, using the equation ST_(e) =R·g·C_(G3) +(v-1)·C_(G3) +Y, where ##EQU23## are the start times returned by the later level invocation of SCHEDULE₋₋ TASKS₋₋ 1 in step 845. Note that the value C_(G3) remains the same through every later invocation of SCHEDULE₋₋ TASKS₋₋ 1.

Referring to FIGS. 9a to 9d, there is illustrated an exemplary flowchart of the procedure PARTITION. As shown in FIG. 9a, PARTITION reads data as tasks T_(j) in group G4 and g, in step 900. PARTITION sequentially assigns tasks T_(J) ε G4 to subsets S_(v) according to slack values, which are the differences between the gcd(P_(j), . . . , P_(n))'s and sums of the computation times C_(j). To be scheduleable, each subset S_(v) must have a non-negative slack value, i.e., greatest common divisor of periods P_(j) in a subset is greater than or equal to the sum of the computation times in the same subset. Thus the goal of PARTITION is to assign tasks T_(j) to the subsets S_(v) such that they all have non-negative slack values. PARTITION attempts to achieve this goal by maximizing the overall slack values for the subsets S_(v). Note that gcd(P_(l), . . . P_(n)) is also referred to herein as gcd(T_(l), . . . , T_(n)).

In steps 905 to 980, PARTITION sequentially assigns to each empty subset S_(v) a task T_(j) from group G4. As will be understood, a subset S_(v) is empty if it has no tasks assigned. The task T_(j) assigned to each empty subset S_(v) is a specific task T_(j) ε G4 that would yield a minimum slack value if that task T_(j) was assigned to any non-empty subsets S₁, . . . , S_(v-1). In other words, PARTITION searches for a combination of task T_(j) and non-empty subset S_(v) which yields the lowest slack value. In step 905, PARTITION assigns a first task T_(j) ε G4 to subset S₁. Specifically, PARTITION examines the slack value for subset S₁ if task T_(j) ε G4 is assigned to subset S₁. A task T_(s) that would yield the least amount of slack if assigned to subset S₁ is subsequently assigned to subset S₁ and removed from group G4 in step 910. In other words, task T_(s) is a task in group G4 where gcd(P_(s))-C_(s) ≦gcd(P_(i))-C_(j) for every task T_(j) εG4. In example 1, task T_(e) is assigned to subset S₁ since it yields a slack of one--tasks T₃, T₅ and T₆ would yield a slack of one, two and two, respectively.

In steps 930, a loop 930 is initiated to assign a task to each of the remaining empty subsets S_(v) one at a time. Note that at the beginning of the loop 930, subsets S₂, . . . , S_(g) are empty. The loop 930 assigns a task T_(j) ε G4 to each of the empty subsets S_(v) according to hypothetical slack values. Specifically, in steps 930 to 965, PARTITION hypothetically assigns task T_(j) to every non-empty subset S_(v), i.e., S₁, . . . , S_(v-1) and stores the slack values for each non-empty subset S_(v) in an array named max[T_(j) ]. In step 935, a loop 935 is initiated to create an array max[T_(j) ] for each task T_(j). In step 940, PARTITION determines the slack value for subset S₁ if the current task T_(j) of the loop 935 was assigned to it. Such slack value is stored in the array max[T_(j) ]. In step 950, as shown in FIG. 9b, a loop 950 is initiated to determine the hypothetical slack values for every other non-empty subset S₂, . . . , S_(v-1) if the current task T_(j) is assigned to them. Such slack values are also stored in their respective max[T_(j) ] arrays.

Upon completing the loop 935 for a current task T_(j), PARTITION proceeds from step 965 to step 970, as shown in FIG. 9c, where PARTITION compares the max[T_(j) ] arrays and finds a task T_(u) in the group G4 such that max[T_(u) ]≦max[T_(j) ] for every T_(j) ε G4. In other words, PARTITION searches through the max[T_(j) ] arrays for the minimum possible slack value. Subsequently, task T_(u) is assigned to the current empty subset S_(v) of the loop 930 and removed from group G4, in step 975. The loop 930 is repeated until no empty subsets S_(v) remain. In example 1, task T₅ is assigned to subset S₂ because its slack value, if assigned to the subset S₁, would be negative one--tasks T₃ and T₆ would yield slack values of zero and negative one if assigned to the non-empty subset S₁.

Upon completing the loop 930, the remaining unassigned tasks T_(j) ε G4 are assigned to subsets S_(v), in steps 982 to 992, such that the overall slack values for the subsets S_(v) are maximized. A loop 982 is initiated, in step 982, for every T_(j) ε G4. The loop 982, in steps 984 to 988, as shown in FIG. 9d, hypothetically assigns the current task T_(j) of the loop 982 to each non-empty subset S_(v) while determining the slack value for each subset S_(v). In step 990, task T_(j) is assigned to the subset S_(v) for which the maximum slack value is yielded. Steps 982 to 992 are repeated until every task T_(j) have been assigned. Upon assigning every task T_(j), in step 994, data indicating the tasks T_(j) which comprises each subset S_(v) is returned to the calling program.

Applying steps 982 to 992 to example 1, task T₃ is assigned to subset S₁ since it would yield slack values of zero and one if task T₃ was assigned to subsets S₁ and S₂, respectively, and task T₆ is assigned to S₂ since it would yield slack values of negative two and positive one if task T₆ was assigned to subsets S₁ and S₂, respectively. Thus, S₁ ={T₂,T₃ } and S₂ ={T₅,T₆ }. Control is returned to the calling program SCHEDULE₋₋ TASKS₋₋ 1 at step 840. Accordingly, a second level invocation of SCHEDULE₋₋ TASKS₋₋ 1 is performed for subsets S₁ ={T₂,T₃ } and S₂ ={T₅,T₆ }. In these second level invocations, the instructions in steps 805 to 824 of SCHEDULE₋₋ TASKS₋₋ 2 determine the subsets S₁ and S₂ to be scheduleable, and start times ST₂ =0, ST₃ =1, ST₅ =0 and ST₆ =1 are returned to the first level invocation of SCHEDULE₋₋ TASKS₋₋ 1 at step 850. The start times ST_(e) are subsequently merged in step 860 and returned to SCHEME-1 at step 680. Merging the start times ST_(e) yields the following start times ST_(e) : ST₂ =0, ST₃ =2, ST₅ =1 and ST₆ =3.

Second Scheduling Scheme

The second scheduling scheme addresses the scheduling problem presented by the first CGS scheme in a video server employing the EPPV paradigm. Specifically, the second scheduling scheme provides a valid method for scheduling non pre-emptible tasks T_(i) having periods P_(i) on p number of processors, wherein the tasks T_(i) comprises w_(i) number of sub-tasks, each of unit computation times, at intervals of F rounds apart.

The second scheduling scheme utilizes four procedures to address the scheduling problem presented by the first CGS scheme. The first procedure, referred to herein as SCHEME₋₋ 2, splits a group G of tasks contained in a service list into two sub-groups G1 and G2 of tasks according to whether the tasks are scheduleable on a single processor. SCHEME₋₋ 2 subsequently attempts to schedule both sub-groups G1 and G2 separately. For the sub-group of tasks not scheduleable on a single processor, i.e., sub-group G1, SCHEME₋₋ 2 schedules such tasks on p' number of processors. For the sub-group of tasks scheduleable on a single processor, i.e., sub-group G2, SCHEME₋₋ 2 calls upon PARTITION to further split the sub-group G2 into subsets G2-y such that there is one subset G2-y for each of the remaining p" number of processors. For each subset G2-y, SCHEME₋₋ 2 calls a third procedure, referred to herein as SCHEDULE₋₋ TASKS₋₋ 2, to determine scheduleability. SCHEDULE₋₋ TASKS₋₋ 2 uses a fourth procedure, referred to herein as IS₋₋ SCHEDULEABLE, to assist it in determining scheduleability of a subset G2-y. If IS₋₋ SCHEDULEABLE determines that a subset G2-y is not scheduleable and SCHEDULE₋₋ TASKS₋₋ 2 determines that the same subset G2-y can be further partitioned, then SCHEDULE₋₋ TASKS₋₋ 2 calls upon PARTITION to further divide the subsets G2-y into sub-subsets S_(v). After the subset G2-y has been partitioned, SCHEDULE₋₋ TASKS₋₋ 2 calls itself to re-determine scheduleability. The second scheduling scheme will recursively partition a sub-group, subset, etc. of tasks until the second scheduling scheme determines whether the sub-group, subset, etc. of tasks is scheduleable or not partitionable any further.

Referring to FIG. 10a to 10c, there is illustrated an exemplary flowchart of the procedure SCHEME₋₋ 2 for scheduling videos in a service list for a video server employing a first CGS scheme, i.e., scheduling non pre-emptible tasks T₁, . . . , T_(n) with w_(i) number of sub-tasks, each with unit computation times, at periods _(l), . . . , P_(n) and intervals F. In step 1000, as shown in FIG. 10a, data corresponding to tasks T_(i) in group G and a value of p are read into the RAM buffer memory of the video server. The value of p, as applied to the second scheduling scheme and in contrast to the first scheduling scheme, refers to the number of processors required to service one disk. Data for group G includes tasks T_(i), w_(i) number of sub-tasks with unit computation times, periods P_(i) and intervals F, i.e., {T_(i),w_(i),P_(i),F}. Note that in the first CGS scheme, the interval between any adjacent sub-tasks belonging to the same task is F.

For certain tasks, it may be the case that scheduling a task T_(i) periodically requires some of its sub-tasks to be scheduled at the same time. As a result, it may not be possible to schedule such a task T_(i) on a single processor. Example 2 is provided for illustration purposes. Consider a task T₁ with period P₁ =4 and w_(i) =3 at intervals F=6. If the start time ST₁ =0, then the three sub-tasks for the first stream of T₁ are scheduled at rounds 0, 6, and 12, respectively. Since P₁ =4, the first sub-task for the third stream of T₁ needs to be scheduled again at round 12. Thus, there is a conflict at round 12 and task T₁ cannot be scheduled on a single processor.

In step 1005, the tasks T_(i) are categorized according to whether two or more sub-tasks belonging to the same task T_(i) will conflict, i.e., will be scheduled at the same time. Categorizing of the tasks T_(i) is performed using the following condition: for a sub-task r and a sub-task s belonging to task T_(i), determine whether ((r-1) F) mod P_(i=) ((s-1)·F) mod P_(i), where 1≦r≦w_(i), 1≦s≦w_(i), and r≠s. If this condition is satisfied, sub-tasks r and s will be scheduled at the same time causing a scheduling conflict and thus, task T_(i) is not scheduleable on a single processor. Tasks not scheduleable on a single processor are categorized in group G1. For a task T_(i) to be scheduleable on a single processor, the above condition must not be satisfied for any combination of sub-tasks r and s in task T_(i). Tasks scheduleable on a single processor are categorized in group G2.

In steps 1010 to 1030, SCHEME₋₋ 2 determines the number of processors needed for tasks T_(i) ε G1. In step 1010, the value of p', i.e., number of processors required to schedule all the tasks T_(i) ε G1, is initially set to zero. In step 1015, a loop 1015 is initiated which calculates the p_(i) number of processors required to schedule each task T_(i) ε G1. In step 1020, the value p_(i) for the current task T_(i) in the loop 1015 is calculated. Specifically, the value p_(i) depends on the value most frequently yielded by the equation ((r-1)·F) mod P_(i) for every sub-task in task T_(i). The frequency of the most frequently yielded value (by the above equation) for a task T_(i) ε G1 is the number of processors required for scheduling the task T_(i). In steps 1025 and 1026, the current value p_(i) is added to p' and the start time for the current task T_(i) is set to zero. Applying the equation ((r-1)·F) mod P_(i) to example 2, the following values are determined for the three sub-tasks, i.e., r=1, 2 and 3, respectively: 0, 2 and 0. Since the value zero is yielded most frequently for task T₂, then the number of processors required for task T₂ is equal to the frequency of the value zero, which is two. The value p_(i) is progressively added to p', in step 1025, for every T_(i) ε G1 and start times ST_(i) for tasks T_(i) ε G1 are set to zero, in step 1026.

As shown in FIG. 10b, upon completing the loop 1015 or if there are no tasks T_(i) ε G1, SCHEME₋₋ 2 proceeds to step 1040 where it determines the number of processors available for scheduling tasks T_(i) ε G2, i.e., the value of p". In step 1045, if the value of p" is less than zero, then there are not enough processors to schedule tasks T_(i) ε G1 and SCHEME₋₋ 2 goes to step 1046 where a cannot₋₋ be₋₋ scheduled statement is returned before going to step 1080 and terminating. Otherwise SCHEME₋₋ 2 goes to step 1050 where it checks whether there are any tasks T_(i) ε G2 to schedule. If there are no tasks T_(i) ε G2, then SCHEME₋₋ 2 goes to step 1051 where it is instructed to go to step 1080 and stop. Otherwise, SCHEME₋₋ G2 proceeds to step 1055 where it checks whether there are any processors available for scheduling T_(i) εG2. If p"=0, then there are no processors available for servicing the tasks T_(i) ε G2, and SCHEME₋₋ 2 goes to step 1056 where a cannot₋₋ be₋₋ scheduled statement is returned before going to step 1080 and terminating. Otherwise, SCHEME₋₋ 2 proceeds to step 1060 where it determines whether to partition the sub-group G2 into subsets G2-y. If no partitioning is required, i.e., p"=1, SCHEME₋₋ 2 sets subset G2-1 equal to sub-group G2 in step 1061, as shown in FIG. 10c. Otherwise, p" >0 and SCHEME₋₋ 2 calls PARTITION to partition the tasks T_(i) ε G2 into p" number of disjoint subsets G2-y, where y=1, . . . , p", in steps 1065 to 1070. Once the tasks T_(i) ε G2 have been partitioned, SCHEME₋₋ 2 calls SCHEDULE₋₋ TASKS₋₋ 2 for each subset G2-y, in step 1075, where the second scheduling scheme determines whether the tasks T_(i) ε 2 are scheduleable on p" number of processors.

Referring to FIGS. 11a to 11b, there is illustrated an exemplary flowchart of SCHEDULE₋₋ TASKS₋₋ 2. As shown in FIG. 11a, in step 1100, SCHEDULE₋₋ TASKS₋₋ 2 reads data as tasks T_(h) belonging to group G3 and a value of F. In step 1101, SCHEDULE₋₋ TASKS₋₋ 2 calls the procedure IS₋₋ SCHEDULEABLE to determine whether tasks T_(h) in group G3 are scheduleable. In step 1105, SCHEDULE₋₋ TASKS₋₋ 2 checks if a cannot₋₋ be₋₋ scheduled statement or a set of start times was returned by the call to IS₋₋ SCHEDULEABLE. If a set of start times was returned, then SCHEDULE₋₋ TASKS₋₋ 2 returns the set of start times to the calling program, in step 1106. Otherwise SCHEDULE₋₋ TASKS₋₋ 2 proceeds to steps 1110 to 1135 where SCHEDULE₋₋ TASKS₋₋ 2 recursively partitions the group G3. In step 1110, SCHEDULE₋₋ TASKS₋₋ 2 sets the value of g equal to the greatest common divisor of the periods P_(h) in the group G3. In steps 1115 to 1135, as shown in FIGS. 11a and 11b, SCHEDULE₋₋ TASKS₋₋ 2 determines whether the group G3 is partitionable, i.e. is g=1, and if group G3 is partitionable, i.e., g>1, SCHEDULE₋₋ TASKS₋₋ 2 recursively partitions the group G3 into subsets S_(v), where v=1, . . . , g, before invoking itself for each subset S_(v). Note that the data ##EQU24## in step 1120. The steps 1115 to 1135 are similar to the steps 825 to 845 belonging to SCHEDULE₋₋ TASKS₋₋ 1, as shown in FIGS. 8b and 8c.

In steps 1140 to 1145, SCHEDULE₋₋ TASKS₋₋ 2 checks the data returned by the succeeding level invocation of SCHEDULE₋₋ TASKS₋₋ 2 in step 1135. If a cannot₋₋ be₋₋ scheduled statement was returned by any succeeding level invocation of SCHEDULE₋₋ TASKS₋₋ 2, the current invocation of SCHEDULE₋₋ TASKS₋₋ 2 returns a cannot₋₋ be₋₋ scheduled statement to its calling program, in step 1141. Otherwise, the current invocation of SCHEDULE₋₋ TASKS₋₋ 2 assumes a set of start times for the tasks T_(h) in each subset S_(v) were returned by the succeeding level invocation of SCHEDULE₋₋ TASKS₋₋ 2. Accordingly, the current invocation of SCHEDULE₋₋ TASKS₋₋ 2, in step 1145, merges the sets of start times returned by the succeeding level invocations of SCHEDULE₋₋ TASKS₋₋ 2 to obtain start times for the tasks T_(h) ε G3. The following equation is used to merge the sets of start times: ST_(h) =(ST_(h) ·g)+(v-1), where ST_(h) is the start time for task T_(h) ε S_(v) and 1≦v≦g. The merged start times for the tasks T_(h) are subsequently returned to the calling program.

Referring to FIGS. 12a to 12c, there is illustrated an exemplary flowchart of the procedure IS₋₋ SCHEDULEABLE which determines scheduleability by creating an array of time slots and attempting to assign sub-tasks to the time slots such that there are no conflicts. As shown in FIG. 12a, in step 1200, IS₋₋ SCHEDULEABLE reads data as tasks T_(h) belonging to group G4 and a value of F. In step 1201, IS₋₋ SCHEDULEABLE creates an array having g number of time slots numbered 0 to g-1, where the value of g is equal to the greatest common divisor of the periods in group G4. In step 1205, each time slot is marked free, i.e., the time slots have no sub-tasks assigned to them. In steps 1210 to 1250, IS₋₋ SCHEDULEABLE attempts to assign each task T_(h) and its sub-tasks to the array of time slots such that the following two conditions hold: (1) if a task is assigned to time slot j, then the next sub-task of the task (if it exists) is assigned to time slot (j+F) mod g; and (2) no two sub-tasks belong to distinct tasks are assigned to the same time slot. An assignment of sub-tasks to time slots that satisfies the above two conditions is a valid assignment.

In step 1210, the tasks T_(h) ε G4 are sorted in descending order according to the w_(h) number of sub-tasks. In step 1215, a loop 1215 is initiated for each task T_(h) ε G4. Specifically, the loop 1215 assigns the sub-tasks r for each task T_(h) ε G2-y, individually in sorted order, to the time slots j of the array created in step 1201, where j=0, . . . , g-1. In step 1220, the loop 1215 begins by setting the value of j equal to zero so that the time slot 0 is the first time slot to be assigned a sub-task r. In step 1225, a loop 1225 is initiated for every time slot j<g. Specifically, the loop 1225 progressively checks if time slots (j+k·F) mod g, for every sub-task r belonging to the current task T_(h) of the loop 1215, are free to be assigned the sub-task r, where k=r-1, i.e., k=0, . . . , w_(h) -1.

As shown in FIG. 12b, in step 1230, the loop 1225 begins by setting k equal to zero so that the first sub-task r of the first task T_(h) in the sorted order, i.e., task with the largest number of sub-tasks, is assigned to the first time slot 0. In steps 1235 and 1240, a loop 1235 is initiated for every sub-task r (except the first sub-task), i.e., for k<w_(h) -1, to determine whether the sub-tasks r belonging to the current task T_(h) can be assigned to time slots (j+k·F) mod g for the current value of j in the loop 1225. If IS₋₋ SCHEDULEABLE determines that the current sub-task r in the loop 1235 can be successfully assigned to the time slot (j+k·F) mod g for the current value of j in step 1240, then IS₋₋ SCHEDULEABLE re-performs the loop 1235 for the next sub-task r with the same value of j. On the other hand, if IS₋₋ SCHEDULEABLE determines that the current sub-task r in the loop 1235 cannot be assigned to the time slot (j+k·F) mod g, in step 1240, then IS₋₋ SCHEDULEABLE exits the loop 1235 and goes to step 1245 where the current value of j is updated to j+1. Subsequently, in step 1250, if IS₋₋ SCHEDULEABLE determines the updated current value of j, i.e., j+1, is equal to the value of g, then the loop 1225 is terminated and a cannot₋₋ be₋₋ scheduled statement is returned to the calling program in step 1255. Otherwise, IS₋₋ SCHEDULEABLE goes to step 1225 from step 1250 to determine whether the time slots (j+k·F) mod g for the updated current value of j are free to be assigned sub-tasks r belonging to the current task T_(h) in the loop 1215.

If the loop 1235 is successfully completed for all the sub-tasks r in the current task T_(h) of the loop 1215, then IS₋₋ SCHEDULEABLE has determined that the corresponding time slots (j+k·F) mod g are free for the current value of j. Accordingly, IS₋₋ SCHEDULEABLE goes to step 1260 from step 1235 where the sub-tasks r for the current task T_(h) are assigned to the time slots (j+k·f) mod g. Subsequently, IS₋₋ SCHEDULEABLE is instructed in step 1265 to re-perform the loop 1215 for the next task T_(h) in the group G4. Upon completing the loop 1215 for every task T_(h) ε G4 without returning a cannot₋₋ be₋₋ scheduled statement, IS₋₋ SCHEDULEABLE goes to step 1270 and returns to SCHEDULE₋₋ TASKS₋₋ 2 the start times for the first sub-tasks of each task T_(h) ε G4, in step 1270.

The above description is an exemplary mode of carrying out the present invention. Although the present invention has been described with reference to specific examples and embodiments for periodically scheduling non pre-emptible tasks on video servers, it should not be construed to limit the present invention in any manner to video servers, and is merely provided for the purpose of describing the general principles of the present invention. It will be apparent to one of ordinary skill in the art that the present invention may be practiced through other embodiments. 

We claim:
 1. A method for scheduling a group of periodically recurring, non pre-emptible tasks on a server, wherein said tasks have predetermined periods P and include w number of sub-tasks separated by intervals F, said group of tasks including a first class of tasks which are scheduleable on single processors, said server having a first sub-group of processors available for processing said first class of tasks, said method comprising the steps of:partitioning said first class of tasks into one or more disjoint sets; and determining scheduleability of said first class of tasks on said first sub-group of processors based on a function of periods for said first class of tasks and a greatest common divisor of said periods P for tasks in said first class of tasks wherein said first class of tasks is scheduleable if two or more sub-tasks belonging to distinct tasks will not be assigned to a same time slot.
 2. The method of claim 1 wherein said step of determining scheduleability of said first class of tasks includes the step of:determining scheduleability of a disjoint set within a division of g number of time slots, wherein g represents a greatest common divisor of periods P for tasks in said disjoint set.
 3. The method of claim 2 wherein said step of determining scheduleability of said disjoint set includes:determining whether a time slot within said division of g number of time slots is free to be assigned a sub-task, wherein said time slot is free if said time slot has no sub-tasks assigned.
 4. The method of claim 2 wherein said step of determining scheduleability of said disjoint set includes the step of:assigning sub-tasks r of a first task in a disjoint set to free time slots (j'+F·k) mod g in said division of g number of time slots, where r=1, . . . , w, k=r-1 and j' is a predetermined value between 0 and g-1.
 5. The method of claim 4 wherein said first task is a task in said disjoint set having a maximum number of sub-tasks r.
 6. The method of claim 4 wherein said step of determining scheduleability of said disjoint set includes the step of:determining whether time slots (j+F·k) mod g are free to be assigned sub-tasks r of another task in said disjoint set for which said sub-tasks r are unassigned, wherein j is a value between 0 and g-1 other than said predetermined value j'.
 7. The method of claim 6 wherein said step of determining scheduleability of said disjoint set includes the step of:assigning said sub-tasks r of another said task in said disjoint set to said times slots (j+F·k) mod g if said time slots (j+F·k) mod g are free.
 8. The method of claim 7 wherein said step of determining scheduleability of said disjoint set includes the step of:repeating said steps of determining whether said time slots (j+F·k) mod g are free and assigning said sub-tasks r of another said task until no unassigned sub-tasks r remain.
 9. The method of claim 7 wherein said step of determining scheduleability of said disjoint set includes the step of:sorting said tasks in said disjoint set to create a sort order according to w number of sub-tasks r.
 10. The method of claim 9 wherein said sub-tasks r of said tasks in said disjoint set are assigned to said time slots (j+k·F) mod g according to said sort order.
 11. The method of claim 7 comprising the additional step of:scheduling said sub-tasks r using said time slots to which said sub-tasks r were assigned.
 12. The method of claim 11 comprising the additional step of:merging schedules of said sub-tasks to obtain a schedule for said tasks in said first class of tasks.
 13. The method of claim 6 wherein said step of determining scheduleability of said disjoint set includes the step of:partitioning said disjoint set of tasks with periods P and sub-tasks at intervals F into g number of disjoint subsets of tasks with periods P/g and sub-tasks at intervals F/g if g is greater than one and said time slots (j+F·k) mod g are not free.
 14. The method of claim 1 comprising the additional step of:determining whether a task is scheduleable on a single processor, wherein a task is scheduleable on a single processor if two or more sub-tasks belonging to said task will not be assigned to a same time slot.
 15. The method of claim 1 wherein said group of tasks includes a second class of tasks which are not scheduleable on a single processor, wherein a task is not scheduleable on a single processor if two or more sub-tasks belonging to said task will be assigned to a same time slot.
 16. The method of claim 15 comprising the additional step of:calculating a number of processors required by said second class of tasks using a value yielded most frequently by equation ((r-1)·F) mod P for each sub-task r.
 17. The method of claim 15 comprising the additional step of:scheduling a predetermined start time for each task in said second class of tasks.
 18. The method of claim 1 wherein said tasks have computation times and said step of partitioning said first class of tasks includes the step of:assigning said tasks to said disjoint sets such that slack values for said disjoint sets are maximized, wherein a slack value is determined by subtracting a sum of said computation times in a disjoint set from a greatest common divisor of said periods in said disjoint set.
 19. The method of claim 1 wherein said step of partitioning said first class of tasks includes the steps of:assigning a first task to a first empty disjoint set, wherein said first task is a task in said first class of tasks which yields a slack value that is less than or equal to any slack value for any other task in said first class of tasks if assigned to said first empty disjoint set, said slack value determined by subtracting a sum of said computation times in a disjoint set from a greatest common divisor of said periods in said disjoint set; assigning an unassigned task in said first class of tasks to an empty disjoint set such that said unassigned task yields a slack value less than or equal to any slack value for any unassigned task in said first class of tasks if assigned to any non-empty disjoint set; repeating said step of assigning said unassigned task until no empty disjoint sets remain; designating an unassigned or non-designated task in said first class of tasks to a non-empty disjoint set such that said non-empty disjoint set to which said unassigned or non-designated task is designated is a disjoint set which yields a maximum slack value if said unassigned or non-designated task is designated to any other non-empty disjoint set; and repeating said step of designating an unassigned or non-designated task until every task in said first class of tasks has been assigned or designated.
 20. The method of claim 1 wherein said first sub-group of processors have p" number of processors and said first class of tasks is partitioned into p" or less number of disjoint sets if p" is greater than one.
 21. The method of claim 1 wherein said server is a video server for servicing tasks corresponding to videos.
 22. A server for programming periodically recurring tasks, said tasks having periods P_(i) and having w number of sub-tasks separated by intervals F, said server having a first sub-group of processors available for processing said first class of tasks said server comprising:means for partitioning a first class of tasks into one or more disjoint sets, wherein said partitioning is performed based on said periods and said first class of tasks include tasks which are scheduleable on single processors; and means for determining scheduleability of said first class of tasks on said first group of processors by ascertaining whether two or more sub-tasks belonging to distinct tasks will be assigned to a same time slot using a greatest common divisor of periods P for said tasks.
 23. The server of claim 22 wherein said means for determining scheduleability of said first class of tasks includes:means for determining scheduleability of a disjoint set within a division of g number of time slots, wherein g represents a greatest common divisor of periods P for tasks in said disjoint set.
 24. The server of claim 23 wherein said means for determining scheduleability of said disjoint set includes:means for assigning sub-tasks r of a first task in a disjoint set to free time slots (j'+F·k) mod g in said division of g number of time slots, where r=1, . . . , w,k=r-1 and j' is a predetermined value between 0 and g-1.
 25. The server of claim 24 wherein said means for determining scheduleability of said disjoint set includes:means for determining whether time slots (j+F·k) mod g are free to be assigned sub-tasks r of another task in said disjoint set for which said sub-tasks r are unassigned, wherein j is a value between 0 and g-1 other than said predetermined value j'.
 26. The server of claim 22 further comprising:means for determining whether a task is scheduleable on a single processor, wherein a task is scheduleable on a single processor if two or more sub-tasks belonging to said task will not be assigned to a same time slot.
 27. The server of claim 26 further comprising:means for calculating a number of processors required by a task in a second class of tasks using a value yielded most frequently by equation ((r-1)·F) mod P for each sub-task r, wherein said second class of tasks includes tasks which are not scheduleable on a single processor.
 28. The server of claim 22 wherein said means for partitioning said first class of tasks includes:means for assigning said tasks to said disjoint sets such that slack values for said disjoint sets are maximized, wherein a slack value is determined by subtracting a sum of computation times in a disjoint set from a greatest common divisor of said periods in said disjoint set.
 29. The server of claim 22 wherein said server is a video server for servicing tasks corresponding to video programming. 